How to Find Exponents That Are Variables


The most surefire way to find exponents that are variables is to use logarithms. Other algebraic methods exist, but they only work for certain types of problems; additionally, such methods tend to take more time. Since logarithms are essentially the inverse of exponentials, it makes sense to use them to find exponential variables. Some background knowledge is required to complete this task; for instance, you must be familiar with terms like base, exponent and variable.

Things You'll Need

  • Calculator that can perform logarithms
  • Pencil
  • Paper
  • Ensure your equation is written such that the base and exponential variable appear on the left side of the equals sign and the single number appears to the right of the equals sign. For instance, rewrite 46 = 5^x so that it reads 5^x = 46. Though this step doesn't affect the solution, it is helpful for explanatory purposes.

  • Take the logarithm of the number on the right side of the equals sign. To do this, press the "log" key on your calculator, followed by the number. Hit the "enter" or "equals" button. Chances are the resultant value is a decimal; if so, round it to the fourth digit. Note this value on your paper. In the case of 5^x = 46, you would find the logarithm of 46, which after rounding is 1.6628.

  • Find the logarithm of the exponent's base, which you've written on the left side of the equals sign. Hit the "log" button on your calculator and then type in the number. Press the "enter" or "equals" key. Again, the resultant value is likely a decimal; if so, round it to the fourth digit. Write this number on your paper. In the previous example, you would take the logarithm of 5, which after rounding produces 0.6989.

  • Divide the result of step 2 by the result of step 3 on your calculator. Essentially, you are dividing the logarithm of the number on the right side of the equation by the logarithm of the base number on the left side of the equation. Round to the digit specified in your individual instructions. In the example, divide 1.6628 by 0.6989. For the sake of consistency, round to the fourth digit, resulting in 2.3792. This is the solution -- the exponential variable, x, equals approximately 2.3792.

  • Check your work by substituting your solution for the exponential variable back into the original equation. Type the base, followed by the "^" symbol and then type the result of step 4. Press the "enter" or "equals" key. If your solution is correct, both sides of the equals sign will reflect approximately the same number. In the case of 5^x = 46, you would perform 5^2.3792, resulting in 46.0245 = 46. Even though the numbers don't match exactly, they differ by only a fractional amount -- due to the previous rounding -- therefore the solution of 2.3792 is correct. If the numbers on each side of the equals sign differ by a significant amount, typically more than 1, then your solution is incorrect; examine your work for errors.

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